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Chapter 19: Problem 12

Following Example Problem \(19.5,\) show that the \(J=1 \rightarrow J=2\) rotational transition is allowed.

Short Answer

Expert verified
The selection rule for pure rotational transitions is \(\Delta J = \pm 1\). For the given transition from J=1 to J=2, we have \(\Delta J = 2 - 1 = 1\), which satisfies the selection rule. Hence, the J=1 to J=2 rotational transition is allowed.

Step by step solution

01

Selection Rules for Rotational Transitions

In order to determine whether a rotational transition is allowed or not, we need to consider the selection rules that govern these transitions. The key selection rule for a pure rotational transition is given by: \[\Delta J = \pm 1\] This means that a rotational transition is allowed if the change in the rotational quantum number J is either +1 or -1.
02

Given Rotational Transition

In this problem, we are given a transition from J=1 to J=2. We can calculate the change in the rotational quantum number J as follows: \[\Delta J = J_{final} - J_{initial} = 2 - 1 = 1\]
03

Verifying the Selection Rule

Now that we have calculated the change in the rotational quantum number J for the given transition, we need to check if it satisfies the selection rule. Since the selection rule states that ΔJ = ±1, our calculated value of ΔJ = 1 indicates that the J=1 to J=2 rotational transition is allowed according to the selection rule. In conclusion, the J=1 to J=2 rotational transition is allowed based on the selection rule for pure rotational transitions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotational Transitions
Rotational transitions refer to the quantum leaps that molecules undergo when they move between different rotational energy levels. These transitions are an essential concept in quantum mechanics, specifically in the study of molecular spectroscopy. Rotational transitions are based on the idea that when a molecule rotates, it can shift from one energy state, characterized by the rotational quantum number \(J\), to another. The molecule absorbs or emits energy during this process, which can be detected and studied in a laboratory setting. Such transitions are often depicted in spectroscopic techniques where different regions of the electromagnetic spectrum can be used to observe and measure these energy changes. Understanding rotational transitions helps scientists gain insight into the structural and dynamic properties of molecules, such as bond lengths and moments of inertia. In simple terms, rotational transitions are like a dancer spinning from one step to another, requiring energy changes that can reveal more about the dance - that is, the molecule's behavior.
Selection Rules
Selection rules are the guiding principles in quantum mechanics that determine whether particular transitions between energy levels are allowed or forbidden. They are crucial for predicting the outcome of spectroscopic experiments. For rotational transitions, the main selection rule is that the change in the rotational quantum number \( \Delta J \) must be \( \pm 1 \). This rule arises because of the conservation of angular momentum, dictating how the rotational state of a molecule should change when it interacts with electromagnetic radiation. Specifically:
  • \( \Delta J = +1 \) corresponds to the absorption of energy, causing the molecule to go to a higher rotational level.
  • \( \Delta J = -1 \) corresponds to the emission of energy, leading the molecule to a lower rotational level.
These selection rules help scientists verify if a particular rotational transition can occur, guiding the analysis of molecular spectra.
Rotational Quantum Number
The rotational quantum number \( J \) is a fundamental part of quantum mechanics which defines the rotational energy states of a molecule. It is an integer that signifies the total angular momentum due to rotation. In essence, \( J \) can be thought of as the number of rotational quanta or units of angular momentum a molecule possesses. It helps in labeling the distinct rotational states that a molecule may occupy. Mathematically, the energy associated with a specific rotational level is determined using\[E_J = \frac{h^2}{8\pi^2I}J(J+1)\]where \( h \) is Planck's constant and \( I \) is the moment of inertia of the rotating molecule. As \( J \) increases, the energy levels become more densely packed. During rotational transitions, the value of \( J \) changes, often by \( \pm 1 \), which is crucial for determining allowed transitions under the selection rules. Understanding this quantum number helps in predicting how molecules behave under rotational motion.

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Most popular questions from this chapter

Overtone transitions in vibrational absorption spectra for which \(\Delta n=+2,+3, \ldots\) are forbidden for the harmonic potential \(V=(1 / 2) k x^{2}\) because \(\mu_{x}^{m n}=0\) for \(|m-n| \neq 1\) as shown in Section \(19.4 .\) However, overtone transitions are allowed for the more realistic anharmonic potential. In this problem, you will explore how the selection rule is modified by including anharmonic terms in the potential. We do so in an indirect manner by including additional terms in the expansion of the dipole moment \(\mu_{x}\left(x_{e}+x\right)=\mu_{0 x}+x\left(d \mu_{x} / d x\right)_{r_{e}}+\ldots\) but assuming that the harmonic oscillator total energy eigenfunctions are still valid. This approximation is valid if the anharmonic correction to the harmonic potential is small. You will show that including the next term in the expansion of the dipole moment, which is proportional to \(x^{2},\) makes the transitions \(\Delta n=\pm 2\) allowed. a. Show that Equation (19.8) becomes $$\begin{array}{l} \mu_{x}^{m 0}=A_{m} A_{0} \mu_{0 x} \int_{-\infty}^{\infty} H_{m}\left(\alpha^{1 / 2} x\right) H_{0}\left(\alpha^{1 / 2} x\right) e^{-\alpha x^{2}} d x \\ \quad+A_{m} A_{0}\left(\frac{d \mu_{x}}{d x}\right)_{x=0} \int_{-\infty}^{\infty} H_{m}\left(\alpha^{1 / 2} x\right) x H_{0}\left(\alpha^{1 / 2} x\right) e^{-\alpha x^{2}} d x \\ \quad+\frac{A_{m} A_{0}}{2 !}\left(\frac{d^{2} \mu_{x}}{d x^{2}}\right)_{x=0} \int_{-\infty}^{\infty} H_{m}\left(\alpha^{1 / 2} x\right) x^{2} H_{0}\left(\alpha^{1 / 2} x\right) e^{-\alpha x^{2}} d x \end{array}$$ b. Evaluate the effect of adding the additional term to \(\mu_{x}^{m n} .\) You will need the recursion relationship \\[ \alpha^{1 / 2} x H_{n}\left(\alpha^{1 / 2} x\right)=n H_{n-1}\left(\alpha^{1 / 2} x\right)+\frac{1}{2} H_{n+1}\left(\alpha^{1 / 2} x\right) \\] c. Show that both the transitions \(n=0 \rightarrow n=1\) and \(n=0 \rightarrow n=2\) are allowed in this case.

A measurement of the vibrational energy levels of \(^{12} \mathrm{C}^{16} \mathrm{O}\) gives the relationship \\[ \widetilde{\nu}(n)=2170.21\left(n+\frac{1}{2}\right) \mathrm{cm}^{-1}-13.461\left(n+\frac{1}{2}\right)^{2} \mathrm{cm}^{-1} \\] where \(n\) is the vibrational quantum number. The fundamental vibrational frequency is \(\tilde{\nu}_{0}=2170.21 \mathrm{cm}^{-1} .\) From these data, calculate the depth \(D_{e}\) of the Morse potential for \(^{12} \mathrm{C}^{16} \mathrm{O}\) Calculate the bond energy of the molecule.

The rotational constant for \(^{14} \mathrm{N}_{2}\) determined from microwave spectroscopy is \(1.99824 \mathrm{cm}^{-1}\). The atomic mass of \(^{14} \mathrm{N}\) is 14.003074007 amu. Calculate the bond length in \(^{14} \mathrm{N}_{2}\) to the maximum number of significant figures consistent with this information.

Show that the selection rule for the two-dimensional rotor in the dipole approximation is \(\Delta m_{l}=\pm 1 .\) Use \(A_{+\phi} e^{i m_{1} \phi}\) and \(A^{\prime}+\phi e^{i m_{2} \phi}\) for the initial and final states of the rotor and \(\mu \cos \phi\) as the dipole moment element.

The rigid rotor model can be improved by recognizing that in a realistic anharmonic potential, the bond length increases with the vibrational quantum number \(n\). Therefore, the rotational constant depends on \(n,\) and it can be shown that \(B_{n}=B-(n+1 / 2) \alpha,\) where \(B\) is the rigid rotor value. The constant \(\alpha\) can be obtained from experimental spectra. For \\[ ^{1} \mathrm{H}^{81} \mathrm{Br}, B=8.46488 \mathrm{cm}^{-1} \text {and } \alpha=0.23328 \mathrm{cm}^{-1} . \text {Using this } \\] more accurate formula for \(B_{n},\) calculate the bond length for HBr in the ground state and for \(n=3\)
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